I am trying to study the braids generated by periodic orbits of diffeomorphisms of compact surfaces (for example, a punctured disk). The diffeomorphisms are generated by integrating a two-dimensional time dependent ODE on the disk, forward in time. Let us assume that the ODE system (and hence the resulting diffeomorphism) is dependent on one bifurcation parameter.

Also assume that we select some 'distinguished' periodic orbits of this system, and start varying the bifurcation parameter. For each value of this parameter, we can form a braid, where 'world-line' of each periodic orbit (in 2+1 space) is a strand. I am interested in exploring the relationship between the different braids that are formed when we vary the parameter, and especially the behavior near a bifurcation point of any one (or more) periodic points that we selected. Are there any results that shed some light on this ? I have been advised to look for connections with winding number of the periodic orbits, but so far I haven't been able to find any relevant literature.

The motivation comes from the fact that braids formed above encode quite a bit of information, including but not limited to, the topological entropy of the flow. One can obtain lower bounds on topological entropy by invoking the Thurston-Nielsen theorem, and so on.

Thanks for any insight.